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Properties

Label 32.0.15693807744...0000.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{96}\cdot 5^{24}\cdot 7^{16}$
Root discriminant $70.77$
Ramified primes $2, 5, 7$
Class number Not computed
Class group Not computed
Galois group $C_2\times C_4^2$ (as 32T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -128, 0, 16304, 0, -2076688, 0, 264513791, 0, -165613696, 0, 70660512, 0, -25593824, 0, 8473857, 0, -2102272, 0, 455328, 0, -88112, 0, 14591, 0, -1664, 0, 176, 0, -16, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 16*x^30 + 176*x^28 - 1664*x^26 + 14591*x^24 - 88112*x^22 + 455328*x^20 - 2102272*x^18 + 8473857*x^16 - 25593824*x^14 + 70660512*x^12 - 165613696*x^10 + 264513791*x^8 - 2076688*x^6 + 16304*x^4 - 128*x^2 + 1)
 
gp: K = bnfinit(x^32 - 16*x^30 + 176*x^28 - 1664*x^26 + 14591*x^24 - 88112*x^22 + 455328*x^20 - 2102272*x^18 + 8473857*x^16 - 25593824*x^14 + 70660512*x^12 - 165613696*x^10 + 264513791*x^8 - 2076688*x^6 + 16304*x^4 - 128*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{32} - 16 x^{30} + 176 x^{28} - 1664 x^{26} + 14591 x^{24} - 88112 x^{22} + 455328 x^{20} - 2102272 x^{18} + 8473857 x^{16} - 25593824 x^{14} + 70660512 x^{12} - 165613696 x^{10} + 264513791 x^{8} - 2076688 x^{6} + 16304 x^{4} - 128 x^{2} + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $32$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 16]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(156938077449417789520626992646455296000000000000000000000000=2^{96}\cdot 5^{24}\cdot 7^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $70.77$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(560=2^{4}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{560}(1,·)$, $\chi_{560}(517,·)$, $\chi_{560}(391,·)$, $\chi_{560}(393,·)$, $\chi_{560}(267,·)$, $\chi_{560}(13,·)$, $\chi_{560}(279,·)$, $\chi_{560}(281,·)$, $\chi_{560}(547,·)$, $\chi_{560}(293,·)$, $\chi_{560}(167,·)$, $\chi_{560}(169,·)$, $\chi_{560}(43,·)$, $\chi_{560}(559,·)$, $\chi_{560}(181,·)$, $\chi_{560}(57,·)$, $\chi_{560}(447,·)$, $\chi_{560}(449,·)$, $\chi_{560}(323,·)$, $\chi_{560}(69,·)$, $\chi_{560}(461,·)$, $\chi_{560}(337,·)$, $\chi_{560}(211,·)$, $\chi_{560}(349,·)$, $\chi_{560}(223,·)$, $\chi_{560}(99,·)$, $\chi_{560}(491,·)$, $\chi_{560}(237,·)$, $\chi_{560}(111,·)$, $\chi_{560}(113,·)$, $\chi_{560}(503,·)$, $\chi_{560}(379,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{14} - \frac{1}{3} a^{12} - \frac{1}{3} a^{8} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{15} - \frac{1}{3} a^{13} - \frac{1}{3} a^{9} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{93} a^{18} - \frac{1}{93} a^{16} - \frac{10}{31} a^{14} - \frac{1}{93} a^{12} + \frac{32}{93} a^{10} + \frac{5}{93} a^{8} + \frac{26}{93} a^{6} - \frac{26}{93} a^{4} - \frac{12}{31} a^{2} - \frac{26}{93}$, $\frac{1}{93} a^{19} - \frac{1}{93} a^{17} - \frac{10}{31} a^{15} - \frac{1}{93} a^{13} + \frac{32}{93} a^{11} + \frac{5}{93} a^{9} + \frac{26}{93} a^{7} - \frac{26}{93} a^{5} - \frac{12}{31} a^{3} - \frac{26}{93} a$, $\frac{1}{93} a^{20} + \frac{37}{93} a^{10} + \frac{5}{93}$, $\frac{1}{93} a^{21} + \frac{37}{93} a^{11} + \frac{5}{93} a$, $\frac{1}{93} a^{22} + \frac{37}{93} a^{12} + \frac{5}{93} a^{2}$, $\frac{1}{93} a^{23} + \frac{37}{93} a^{13} + \frac{5}{93} a^{3}$, $\frac{1}{93} a^{24} + \frac{37}{93} a^{14} + \frac{5}{93} a^{4}$, $\frac{1}{93} a^{25} + \frac{37}{93} a^{15} + \frac{5}{93} a^{5}$, $\frac{1}{66760355287546563} a^{26} + \frac{172721887392179}{66760355287546563} a^{24} + \frac{23017152462407}{66760355287546563} a^{22} + \frac{1839142177020}{717853282661791} a^{20} - \frac{125845884110966}{66760355287546563} a^{18} + \frac{5235897770280454}{66760355287546563} a^{16} - \frac{8803787873989466}{22253451762515521} a^{14} - \frac{8732779905152626}{22253451762515521} a^{12} - \frac{10147904688940186}{66760355287546563} a^{10} + \frac{7118268598229848}{66760355287546563} a^{8} - \frac{17118585194915102}{66760355287546563} a^{6} + \frac{29848450250669584}{66760355287546563} a^{4} - \frac{6278989835939942}{22253451762515521} a^{2} - \frac{30919609174006885}{66760355287546563}$, $\frac{1}{66760355287546563} a^{27} + \frac{172721887392179}{66760355287546563} a^{25} + \frac{23017152462407}{66760355287546563} a^{23} + \frac{1839142177020}{717853282661791} a^{21} - \frac{125845884110966}{66760355287546563} a^{19} + \frac{5235897770280454}{66760355287546563} a^{17} - \frac{8803787873989466}{22253451762515521} a^{15} - \frac{8732779905152626}{22253451762515521} a^{13} - \frac{10147904688940186}{66760355287546563} a^{11} + \frac{7118268598229848}{66760355287546563} a^{9} - \frac{17118585194915102}{66760355287546563} a^{7} + \frac{29848450250669584}{66760355287546563} a^{5} - \frac{6278989835939942}{22253451762515521} a^{3} - \frac{30919609174006885}{66760355287546563} a$, $\frac{1}{2069571013913943453} a^{28} - \frac{1}{2069571013913943453} a^{26} - \frac{6505439162493755}{2069571013913943453} a^{24} + \frac{3233118337864}{3455043428904747} a^{22} - \frac{127011925929511}{2069571013913943453} a^{20} - \frac{1703350628711584}{689857004637981151} a^{18} + \frac{303089952500312846}{2069571013913943453} a^{16} - \frac{251110944806779610}{689857004637981151} a^{14} - \frac{858996449948479670}{2069571013913943453} a^{12} + \frac{183096079549896622}{689857004637981151} a^{10} - \frac{967396697724619952}{2069571013913943453} a^{8} + \frac{878928924427783577}{2069571013913943453} a^{6} - \frac{477962746632348250}{2069571013913943453} a^{4} + \frac{1021002967383626981}{2069571013913943453} a^{2} - \frac{1021288458245044978}{2069571013913943453}$, $\frac{1}{2069571013913943453} a^{29} - \frac{1}{2069571013913943453} a^{27} - \frac{6505439162493755}{2069571013913943453} a^{25} + \frac{3233118337864}{3455043428904747} a^{23} - \frac{127011925929511}{2069571013913943453} a^{21} - \frac{1703350628711584}{689857004637981151} a^{19} + \frac{303089952500312846}{2069571013913943453} a^{17} - \frac{251110944806779610}{689857004637981151} a^{15} - \frac{858996449948479670}{2069571013913943453} a^{13} + \frac{183096079549896622}{689857004637981151} a^{11} - \frac{967396697724619952}{2069571013913943453} a^{9} + \frac{878928924427783577}{2069571013913943453} a^{7} - \frac{477962746632348250}{2069571013913943453} a^{5} + \frac{1021002967383626981}{2069571013913943453} a^{3} - \frac{1021288458245044978}{2069571013913943453} a$, $\frac{1}{2069571013913943453} a^{30} + \frac{4949579888216462}{2069571013913943453} a^{20} + \frac{318413444528890163}{689857004637981151} a^{10} + \frac{210320381790778352}{2069571013913943453}$, $\frac{1}{2069571013913943453} a^{31} + \frac{4949579888216462}{2069571013913943453} a^{21} + \frac{318413444528890163}{689857004637981151} a^{11} + \frac{210320381790778352}{2069571013913943453} a$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $15$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{41834283536}{689857004637981151} a^{30} - \frac{460177118896}{689857004637981151} a^{28} + \frac{4350765487744}{689857004637981151} a^{26} - \frac{38150251942111}{689857004637981151} a^{24} + \frac{321204605564400}{689857004637981151} a^{22} - \frac{38403872286048}{22253451762515521} a^{20} + \frac{5496690182362112}{689857004637981151} a^{18} - \frac{22156108523844897}{689857004637981151} a^{16} + \frac{66918705624155104}{689857004637981151} a^{14} - \frac{560798999534176}{689857004637981151} a^{12} + \frac{433020644744306816}{689857004637981151} a^{10} - \frac{691609058242265311}{689857004637981151} a^{8} + \frac{5429797162988048}{689857004637981151} a^{6} - \frac{42629134923184}{689857004637981151} a^{4} + \frac{87869093584378842752}{689857004637981151} a^{2} - \frac{2614642721}{689857004637981151} \) (order $10$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_4^2$ (as 32T36):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 32
The 32 conjugacy class representatives for $C_2\times C_4^2$
Character table for $C_2\times C_4^2$ is not computed

Intermediate fields

\(\Q(\sqrt{35}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{70}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{2}, \sqrt{35})\), 4.0.2048.2, 4.0.2508800.1, \(\Q(\sqrt{10}, \sqrt{14})\), \(\Q(\sqrt{5}, \sqrt{7})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{7})\), 4.0.51200.2, 4.0.100352.5, \(\Q(\sqrt{7}, \sqrt{10})\), \(\Q(\sqrt{5}, \sqrt{14})\), 4.4.256000.2, 4.4.12544000.1, 4.4.256000.1, 4.4.12544000.2, \(\Q(\zeta_{5})\), 4.0.98000.1, 4.0.8000.2, 4.0.392000.2, 8.0.25176309760000.40, 8.8.98344960000.1, 8.0.25176309760000.66, 8.0.2621440000.1, 8.0.40282095616.1, 8.0.6294077440000.7, 8.0.25176309760000.41, 8.8.629407744000000.3, 8.8.629407744000000.2, 8.0.9604000000.3, 8.0.2458624000000.3, 8.8.65536000000.1, 8.8.157351936000000.4, 8.0.64000000.2, 8.0.2458624000000.6, 8.8.629407744000000.5, 8.8.629407744000000.4, 8.0.153664000000.6, 8.0.2458624000000.1, 16.0.633846573131471257600000000.9, 16.16.396154108207169536000000000000.2, 16.0.6044831973376000000000000.5, 16.0.4294967296000000000000.2, 16.0.396154108207169536000000000000.13, 16.0.396154108207169536000000000000.6, 16.0.24759631762948096000000000000.4

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.4.0.1}{4} }^{8}$ R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
5Data not computed
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$